For a closed, orientable hyperbolic –manifold and an onto homomorphism that is not induced by a fibration , we bound the ranks of the subgroups for , below, linearly in . The key new ingredient is the following result: if is a closed, orientable hyperbolic –manifold and is a connected, two-sided incompressible surface of genus that is not a fiber or semifiber, then a reduced homotopy in has length at most .
"Explicit rank bounds for cyclic covers." Algebr. Geom. Topol. 16 (3) 1343 - 1371, 2016. https://doi.org/10.2140/agt.2016.16.1343